Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature

TL;DR

This paper establishes rigidity results for invariant Ricci flow blowdown limits on nilpotent principal bundles with zero curvature, classifying the resulting solitons especially for the Heisenberg group and one-dimensional bases.

Contribution

It introduces a new functional to analyze Ricci flows on nilpotent bundles, leading to a complete local classification of blowdown limits in four dimensions.

Findings

Blowdown limits are locally expanding Ricci solitons for the Heisenberg group.

Classification of solitons when the base is one-dimensional.

Complete local classification of invariant Ricci flow blowdown limits in four-dimensional nilpotent bundles.

Abstract

We use Lott's functional and construct a new functional to derive rigidity results for invariant Ricci flow blowdown limits on nilpotent principal bundles with zero associated curvature. Consequently, we prove that the blowdown limit is locally an expanding Ricci soliton when the structure group is the three dimensional Heisenberg group. In addition, we classify this soliton when the base manifold is one dimensional. This, together with Lott's work in the abelian setting, yields a complete local classification of invariant Ricci flow blowdown limits on four dimensional, nilpotent principal bundles.